The Tensor Product Representation of Polynomials of Weak Type in a DF-Space.

Let E and F be locally convex spaces over C and let P(nE; F) be the space of all continuous n-homogeneous polynomials from E to F. We denote by ⊗n,s,πE the n-fold symmetric tensor product space of E endowed with the projective topology. Then, it is well known that each polynomialp ∈ P(nE; F) is represented as an element in the space L(⊗n,s,πE; F) of all continuous linear mappings from ⊗n,s,πE to F. A polynomial p ∈ P(nE; F) is said to be of weak type if, for every bounded set B of E, p|B is weakly continuous on B. We denote by Pw(nE; F) the space of all n-homogeneous polynomials of weak type from E to F. In this paper, in case that E is a DF space, we will give the tensor product representation of the space Pw(nE; F). [ABSTRACT FROM AUTHOR]